1. Field of the Invention
This invention generally relates to power supply networks, and more particularly to a method for determining voltage and current distributions in a power supply network when known voltages and currents are applied.
2. Description of the Related Art
Numerical methods for approximating voltage and current distributions in a power supply network have been proposed. One class of methods are known as finite difference methods. Most often these methods are used to determine voltage and current distributions when contacts and holes in the network have rectangular boundaries. By rectangular boundaries, we mean boundaries composed of straight lines which are either parallel or perpendicular to each other. See, for example, Balabanian et al., Electrical Network Theory, John Wiley and Sons, New York, 1968.
In implementing this method, it is assumed that the network is composed of disjoint sub-regions with distinct, constant resistivities. The entire region is then covered with a rectangular grid of intersecting horizontal and vertical lines. The grid lines are required to include the boundaries of all the holes and contacts in the network, as well as the boundaries between all sub-regions. The voltage values at the grid points are then determined by requiring them to satisfy a sparse linear system of equations arising from a discrete approximation to conservation of current and Ohm""s law. These equations are also disclosed in the aforementioned Balabanian publication. The coefficients of the equations depend on a xe2x80x9cnumerical conductance,xe2x80x9d which in turn depends on the physical resistance near a grid point and the distances between nearby grid points.
The Inventors of the present application have discovered that the standard finite difference method does not produce an accurate voltage distribution approximation when the network includes holes and contacts with non-rectangular boundaries, unless a very fine grid is used to approximate non-rectangular boundaries. This is because the standard finite difference method does not take into account the precise geometry of non-rectangular holes and contacts, such as boundary direction and/or curvature, and are therefore less accurate.
A contact or hole with a non-rectangular boundary can be approximated using a very fine rectangular mesh, however this technique has also proven undesirable. Very fine rectangular meshes, for example, greatly increase the number of mesh lines. This, in turn, increases the number of grid points in the problem, which then increases the size of the problem which must be solved. And, as a result of this increased complexity, greater amounts of computer time are needed to solve the problem.
Finite element methods have also been used for solving resistance problems on regions with non-rectangular boundaries. See, for example, Sakkas, Potential Distribution and Multi-Terminal DC Resistance for LSI Technology, IBM J. Res, Devel., vol. 23, No. 6, page 640, 1979. These methods are not well suited to solving resistance problems, since the problems often contain long narrow holes. Finite element discretizations of such regions would therefore contain triangles with high aspect ratios. The presence of such triangles decreases the accuracy of the finite element method.
A need therefore exists for an improved numerical method of approximating voltage and current distributions in a power supply network, and moreover one which accurately approximates voltage and current distributions in a resistive network which includes holes or contacts with non-rectangular boundaries.
It is one object of the present invention to provide an accurate and efficient method for approximating voltage and current distributions in a power supply network.
It is another object of the present invention to achieve the aforementioned object by solving the resistance problem in a power supply network using a modification of a rectangular grid algorithm which approximates current and voltage distributions in a structure having holes and/or contacts with non-rectangular boundaries. These boundaries may be arbitrary shapes, circles, ellipses, or any other shape that is not rectangular. By non-rectangular, we mean contacts or holes whose boundaries are not composed of straight lines which are parallel to grid lines.
The foregoing and other objects of the invention are achieved by providing a method for approximating voltage, current, and/or power distributions in a power supply network which includes a structure having non-rectangular holes and contact regions where specific voltage or current sources are attached to drive the structure. The method includes parsing a resistive region of the structure into rectangles, applying a grid over the parsed resistive region, locating grid points near non-rectangular holes and contacts, generating standard finite difference equations at points not near the non-rectangular holes or contacts, generating specially modified finite difference equations at points near non-rectangular holes and contacts, solving the resulting linear system of equations, and then deriving voltages at the grid points based on solutions to the standard and modified finite difference equations. Current and power distributions may then be derived from this voltage distribution. The method advantageously may be applied to resistive regions having discontinuous or different material resistivities. As a result this method, a more accurate approximation of these distributions are obtained when the resistive region contains holes or contacts with non-rectangular boundaries than when the finite difference equations are not modified.